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Inspired by the corresponding problem in QCD, we determine the pressure ofmassless O(N) scalar field theory up to order g^6 in the weak-couplingexpansion, where g^2 denotes the quartic coupling constant. This necessitatesthe computation of all 4-loop vacuum graphs at a finite temperature: by makinguse of methods developed by Arnold and Zhai at 3-loop level, we demonstratethat this task is manageable at least if one restricts to computing thelogarithmic terms analytically, while handling the ``constant'' 4-loopcontributions numerically. We also inspect the numerical convergence of theweak-coupling expansion after the inclusion of the new terms. Finally, we pointout that while the present computation introduces strategies that should behelpful for the full 4-loop computation on the QCD-side, it also highlights theneed to develop novel computational techniques, in order to be able to completethis formidable task in a systematic fashion.Comment: 34 page

Four-loop pressure of masslessO(N) scalar field theory